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The constant chord theorem is a statement in elementary geometry about a property of certain chords in two intersecting circles.
The circles k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} intersect in the points P {\displaystyle P} and Q {\displaystyle Q}. Z 1 {\displaystyle Z_{1}} is an arbitrary point on k 1 {\displaystyle k_{1}} being different from P {\displaystyle P} and Q {\displaystyle Q}. The lines Z 1 P {\displaystyle Z_{1}P} and Z 1 Q {\displaystyle Z_{1}Q} intersect the circle k 2 {\displaystyle k_{2}} in P 1 {\displaystyle P_{1}} and Q 1 {\displaystyle Q_{1}}. The constant chord theorem then states that the length of the chord P 1 Q 1 {\displaystyle P_{1}Q_{1}} in k 2 {\displaystyle k_{2}} does not depend on the location of Z 1 {\displaystyle Z_{1}} on k 1 {\displaystyle k_{1}} , in other words the length is constant.
The theorem stays valid when Z 1 {\displaystyle Z_{1}} coincides with P {\displaystyle P} or Q {\displaystyle Q} , provided one replaces the then undefined line Z 1 P {\displaystyle Z_{1}P} or Z 1 Q {\displaystyle Z_{1}Q} by the tangent on k 1 {\displaystyle k_{1}} at Z 1 {\displaystyle Z_{1}}.
A similar theorem exists in three dimensions for the intersection of two spheres. The spheres k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} intersect in the circle k s {\displaystyle k_{s}}. Z 1 {\displaystyle Z_{1}} is arbitrary point on the surface of the first sphere k 1 {\displaystyle k_{1}} , that is not on the intersection circle k s {\displaystyle k_{s}}. The extended cone created by k s {\displaystyle k_{s}} and Z 1 {\displaystyle Z_{1}} intersects the second sphere k 2 {\displaystyle k_{2}} in a circle. The length of the diameter of this circle is constant, that is it does not depend on the location of Z 1 {\displaystyle Z_{1}} on k 1 {\displaystyle k_{1}}.